c09ccf computes the one-dimensional multi-level discrete wavelet transform (DWT). The initialization routine c09aaf must be called first to set up the DWT options.
The routine may be called by the names c09ccf or nagf_wav_dim1_multi_fwd.
3Description
c09ccf computes the multi-level DWT of one-dimensional data. For a given wavelet and end extension method, c09ccf will compute a multi-level transform of a data array,
${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$,
using a specified number, ${n}_{\mathrm{fwd}}$, of levels. The number of levels specified, ${n}_{\mathrm{fwd}}$, must be no more than the value ${l}_{\mathrm{max}}$ returned in nwlmax by the initialization routine c09aaf for the given problem. The transform is returned as a set of coefficients for the different levels (packed into a single array) and a representation of the multi-level structure.
The notation used here assigns level $0$ to the input dataset, $x$, with level $1$ being the first set of coefficients computed, with the detail coefficients, ${d}_{1}$, being stored while the approximation coefficients, ${a}_{1}$, are used as the input to a repeat of the wavelet transform. This process is continued until, at level ${n}_{\mathrm{fwd}}$, both the detail coefficients, ${d}_{{n}_{\mathrm{fwd}}}$, and the approximation coefficients, ${a}_{{n}_{\mathrm{fwd}}}$ are retained. The output array, $C$, stores these sets of coefficients in reverse order, starting with ${a}_{{n}_{\mathrm{fwd}}}$ followed by ${d}_{{n}_{\mathrm{fwd}}},{d}_{{n}_{\mathrm{fwd}}-1},\dots ,{d}_{1}$.
4References
None.
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: the number of elements, $n$, in the data array $x$.
Constraint:
this must be the same as the value n passed to the initialization routine c09aaf.
2: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: x contains the one-dimensional input dataset
${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
3: $\mathbf{lenc}$ – IntegerInput
On entry: the dimension of the array c as declared in the (sub)program from which c09ccf is called. c must be large enough to contain the number, ${n}_{c}$, of wavelet coefficients. The maximum value of ${n}_{c}$ is returned in nwc by the call to the initialization routine c09aaf and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels, ${n}_{\mathrm{fwd}}$, is chosen to be less than the maximum, then ${n}_{c}$ is correspondingly smaller and lenc can be reduced by noting that the number of coefficients at each level is given by $\lceil \overline{n}/2\rceil $ for ${\mathbf{mode}}=\text{'P'}$ in c09aaf and $\lfloor (\overline{n}+{n}_{f}-1)/2\rfloor $ for ${\mathbf{mode}}=\text{'H'}$, $\text{'W'}$ or $\text{'Z'}$, where $\overline{n}$ is the number of input data at that level and ${n}_{f}$ is the filter length provided by the call to c09aaf. At the final level the storage is doubled to contain the set of approximation coefficients.
Constraint:
${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the number of approximation and detail coefficients that correspond to a transform with nwlmax levels.
4: $\mathbf{c}\left({\mathbf{lenc}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: let $q\left(\mathit{i}\right)$ denote the number of coefficients (of each type) produced by the wavelet transform at level $\mathit{i}$, for $\mathit{i}={n}_{\mathrm{fwd}},{n}_{\mathrm{fwd}}-1,\dots ,1$. These values are returned in dwtlev. Setting ${k}_{1}=q\left({n}_{\mathrm{fwd}}\right)$ and
${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q({n}_{\mathrm{fwd}}-\mathit{j}+1)$, for $\mathit{j}=1,2,\dots ,{n}_{\mathrm{fwd}}$, the coefficients are stored as follows:
${\mathbf{c}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{k}_{1}$
Contains the level ${n}_{\mathrm{fwd}}$ approximation coefficients, ${a}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left(\mathit{i}\right)$, for $\mathit{i}={k}_{1}+1,\dots ,{k}_{2}$
Contains the level ${n}_{\mathrm{fwd}}$ detail coefficients ${d}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left(\mathit{i}\right)$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level
${n}_{\mathrm{fwd}}-\mathit{j}+1$ detail coefficients, for $\mathit{j}=2,3,\dots ,{n}_{\mathrm{fwd}}$.
5: $\mathbf{nwl}$ – IntegerInput
On entry: the number of levels, ${n}_{\mathrm{fwd}}$, in the multi-level resolution to be performed.
Constraint:
$1\le {\mathbf{nwl}}\le {l}_{\mathrm{max}}$, where ${l}_{\mathrm{max}}$ is the value returned in nwlmax (the maximum number of levels) by the call to the initialization routine c09aaf.
On exit: the number of transform coefficients at each level.
${\mathbf{dwtlev}}\left(1\right)$ and ${\mathbf{dwtlev}}\left(2\right)$ contain the number, $q\left({n}_{\mathrm{fwd}}\right)$, of approximation and detail coefficients respectively, for the final level of resolution (these are equal); ${\mathbf{dwtlev}}\left(\mathit{i}\right)$ contains the number of detail coefficients, $q({n}_{\mathrm{fwd}}-\mathit{i}+2)$, for the (${n}_{\mathrm{fwd}}-\mathit{i}+2$)th level, for $\mathit{i}=3,4,\dots ,{n}_{\mathrm{fwd}}+1$.
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine c09aaf.
On exit: contains additional information on the computed transform.
8: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, n is inconsistent with the value passed to the initialization routine: ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$, n should be $\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=3$
On entry, lenc is set too small: ${\mathbf{lenc}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lenc}}\ge \u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{nwl}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{nwl}}\ge 1$.
On entry, nwl is larger than the maximum number of levels returned by the initialization routine: ${\mathbf{nwl}}=\u27e8\mathit{\text{value}}\u27e9$, maximum $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=7$
Either the initialization routine has not been called first or array icomm has been corrupted.
Either the initialization routine was called with ${\mathbf{wtrans}}=\text{'S'}$ or array icomm has been corrupted.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The accuracy of the wavelet transform depends only on the floating-point operations used in the convolution and downsampling and should thus be close to machine precision.
8Parallelism and Performance
c09ccf is not threaded in any implementation.
9Further Comments
The wavelet coefficients at each level can be extracted from the output array c using the information contained in dwtlev on exit (see the descriptions of c and dwtlev in Section 5). For example, given an input data set, $x$, denoising can be carried out by applying a thresholding operation to the detail coefficients at every level. The elements ${\mathbf{c}}\left(i\right)$, for $i={k}_{1}+1,\dots ,{k}_{{n}_{\mathrm{fwd}}}+1$, as described in Section 5, contain the detail coefficients, ${\hat{d}}_{\mathit{i}\mathit{j}}$, for $\mathit{i}={n}_{\mathrm{fwd}},{n}_{\mathrm{fwd}}-1,\dots ,1$ and $\mathit{j}=1,2,\dots ,q\left(i\right)$, where ${\hat{d}}_{ij}={d}_{ij}+\sigma {\epsilon}_{ij}$ and $\sigma {\epsilon}_{ij}$ is the transformed noise term. If some threshold parameter $\alpha $ is chosen, a simple hard thresholding rule can be applied as
taking ${\overline{d}}_{ij}$ to be an approximation to the required detail coefficient without noise, ${d}_{ij}$. The resulting coefficients can then be used as input to c09cdf in order to reconstruct the denoised signal.
See the references given in the introduction to this chapter for a more complete account of wavelet denoising and other applications.
10Example
This example performs a multi-level resolution of a dataset using the Daubechies wavelet (see ${\mathbf{wavnam}}=\text{'DB4'}$ in c09aaf) using zero end extensions, the number of levels of resolution, the number of coefficients in each level and the coefficients themselves are reused. The original dataset is then reconstructed using c09cdf.